KLAMATH FALLS, Ore.—The Republican Party’s revamp of the U.S. health-care system is heading toward a showdown over Medicaid.

This region, a patch of red in a largely blue state, saw more changes than most from the 2010 Affordable Care Act’s expansion of the health-insurance program for low-income people. Its uninsured rate fell sharply. Its sole hospital nearly tripled its Medicaid revenue, helping finance an expansion. ACA funds…